Abstract

The cyclic graph of a finite group is
as follows: take as the vertices of and join two distinct
vertices and if is cyclic. In this paper, we
investigate how the graph theoretical properties of affect the group theoretical
properties of . First, we consider some properties of
and characterize certain finite groups whose cyclic graphs have some properties.
Then, we present some properties of the cyclic graphs of
the dihedral groups and the generalized quaternion groups for some .
Finally, we present some parameters about the cyclic graphs of finite noncyclic groups of
order up to .

1. Introduction and Results

Recently, study of algebraic structures by graphs associated with them gives rise to many interesting results. There are many papers on assigning a graph to a group and algebraic properties of group by using the associated graph; for instance, see [1–4].

Let be a group with identity element . One can associate a graph to in many different ways. Abdollahi and Hassanabadi introduced a graph (called the noncyclic graph of a group; see [4]) associated with a group by the cyclicity of subgroups. It is a graph whose vertex set is the set , where is cyclic for all and is adjacent if is not a cyclic subgroup. They established some graph theoretical properties (such as regularity) of this graph in terms of the group ones.

In this paper, we consider the converse. We associate a graph with (called the cyclic graph of ) as follows: take as the vertices of and two distinct vertices and are adjacent if and only if is a cyclic subgroup of . For example, Figure 1 is the cyclic graph of , and Figure 2 is . For any group , it is easy to see that the cyclic graph is simple and undirected with no loops and multiple edges. By the definition, we shall explore how the graph theoretical properties of affect the group theoretical properties of . In particular, the structure of the group by some graph theoretical properties of the associated graph is determined.

Figure 1: .

Figure 2: .

The outline of this paper is as follows. In Section 2, we introduce a lot of basic concepts and notations of group and graph theory which will be used in the sequel. In Section 3, we give some properties of the cyclic graph of a group on diameter, planarity, partition, clique number, and so forth and characterize a finite group whose cyclic graph is complete (planar, a star, regular, etc.). For example, the cyclic graph of any group is always connected whose diameter is at most 2 and the girth is either 3 or ; the cyclic graph of group is complete if and only if is cyclic and is a star if and only if is an elementary abelian 2-group. In particular, for a finite group , if and only if , the Klein group. In Section 4, we present some properties of the cyclic graphs of the dihedral groups , including degrees of vertices, traversability (Eulerian and Hamiltonian), planarity, coloring, and the number of edges and cliques. Furthermore, we get the automorphism group of for all . Particularly, for all , if is a group with , then . Similar to Section 4, we discuss the properties of the cyclic graphs on the generalized quaternion groups in Section 5. In Section 6, we obtain some parameters on the cyclic graphs of finite noncyclic groups of order up to .

2. Preliminaries

In this paper, we consider simple graphs which are undirected, with no loops or multiple edges. Let be a graph. We will denote and the set of vertices and edges of , respectively. is, respectively called empty and complete if is empty and every two distinct vertices in are adjacent. A complete graph of order is denoted by . The degree of a vertex in , denoted by , is the number of edges which are incident to . A subset of is called a clique if the induced subgraph of is complete. The order of the largest clique in is its clique number, which is denoted by . A -vertex coloring of is an assignment of colors to the vertices of such that no two adjacent vertices have the same color. The chromatic number of is the minimum for which has a -vertex coloring. If , then denotes the length of the shortest path between and . The largest distance between all pairs of is called the diameter of and denoted by . The length of the shortest cycle in the graph is called girth of ; if does not contain any cycles, then its girth is defined to be infinity (). For a vertex of , denote by the set of vertices in which are adjacent to . A vertex of is a cutvertex if is disconnected. An path of length is called an geodesic; the closed interval of and is the set of those vertices belonging to at least one geodesic. A set of is called a geodetic set for if , where . A geodetic set of minimum cardinality in is called a minimum geodetic set and this cardinality is the geodetic number. A set of vertices of is a dominating set of if every vertex in is adjacent to some vertex in ; the cardinality of a minimum dominating set is called the domination number of and is denoted by . is a bipartite graph means that can be partitioned into two subsets and , called partite sets, such that every edge of joins a vertex of and a vertex of . If every vertex of is adjacent to every vertex of , is called a complete bipartite graph, where and are independent. A complete bipartite graph with and is denoted by . For more information, the reader can refer to [5].

In this paper, all groups considered are finite. Let be a finite group with identity element . The number of elements of is called its order and is denoted by . The order of an element of is the smallest positive integer such that . The order of an element is denoted by . For more notations and terminologies in group theory consult [6].

3. Some Properties of the Cyclic Graphs

Definition 1. In group theory, a locally cyclic group is a group in which every finitely generalized subgroup is cyclic. A group is locally cyclic if and only if every pair of elements in the group generates a cyclic group. It is a fact that every finitely generalized locally cyclic group is cyclic. So a finite locally cyclic group is cyclic.

Definition 2 (see [7, 8]). Let be a group. The cyclicizer of an element of , denoted , is defined by
In general, the cyclicizer of is not a subgroup of . For example, let , then is not a subgroup.

Definition 3. The cyclicizer of is defined as follows:
By [9, Theorem 1], is a normal subgroup of and .

Definition 4. Let be a group. The cyclic graph of is a graph with and two distinct vertices are adjacent in if and only if is a cyclic subgroup of .

Proposition 6. Let be a group with the identity element . Then . In particular, is connected and the girth of is either or .

Proof. Suppose that and are two distinct vertices of . If is a cyclic subgroup of , then is adjacent to , and hence . Thus we may assume that is not cyclic. Note that both and are cyclic and the vertices and are adjacent to ; hence we get . This means that is connected and . If there exist such that and are joined by some edge, then is a cycle of order 3 of and so the girth of is . Otherwise, every two vertices (nonidentity elements of ) of are not adjacent; that is, is a star, which implies that the girth of is equal to .

The following proposition is obvious; we omit its proof.

Proposition 7. Let be a group with . Then is a dominating set of order of . In particular, and .

Corollary 8. Let be a group. Then is a dominating set if and only if . Moreover, the number of the dominating sets of size 1 is .

Theorem 9. Let be a nontrivial group. Then (or equivalently is complete) if and only if is a cyclic group.

Proof. Let and be two arbitrary elements of . Suppose that . Then is a cyclic subgroup of . By Definition 1, is a cyclic group as is finite. For the converse, if is a cyclic group, then is a cyclic subgroup of . Thus , as desired.

Corollary 10. Let be noncyclic group. Then is not regular.

Theorem 11. Let be a group with the identity element . Then (or equivalently is a star) if and only if is an elementary abelian 2-group.

Proof. Assume that is a star. Let be a nonidentity element of . If , then and are adjacent since is a cyclic subgroup of , which is contrary to being a star. Hence . It follows that the order of every element of is 2. If and are two elements of , then , and hence . It means that is an abelian group and . It follows that is an elementary abelian 2-group.Conversely, suppose that is an elementary abelian 2-group. Then the order of every cyclic subgroup of is 2. Let is a nonidentity element of . If there exists an element such that is cyclic, then , which implies . Note that is an element of order 2; then as . It follows that the unique element is adjacent to in . So .

Corollary 12. Let be an elementary abelian 2-group. Then is isomorphic to the symmetric on letters.

Corollary 13. Let be group. Then is a tree if and only if is an elementary abelian 2-group.

Corollary 14. Let be group with . If is bipartite, then .

Proof. Assume, on the contrary, . Then there exist two adjacent vertices and such that . Since , there is an element such that . By Definition 3, is a cycle of length 3 and so the subgraph of induced by is an odd cycle, which is a contradiction to being bipartite (see [5, Theorem 1.12, page 22]).

Remark 15. Let . Then is a bipartite graph, while .

Corollary 16. Let be group. Then is bipartite if and only if is an elementary abelian 2-group.

Proposition 17. Let and be two groups. If , then .

Proof. Let be an isomorphism from to . Obviously, is a one-to-one correspondence between and . Let and be two vertices of . If is cyclic, then there exist two positive integers such that and , so and ; It means that , that is, is a subgroup of . Thus is cyclic. Note that is invertible. It follows that and are adjacent in if and only if is adjacent to in . Consequently, is a graph automorphism from to , namely, .

Remark 18. The converse of Proposition 17 is not true in general. Let be the modular group of order 16 (a group is called a modular group if its lattice of subgroups is modular) with presentation
Clearly, . Let . For , this is the same subgroup lattice structure as for the lattice of subgroups of . It is easy to see that , however, because is not abelian.

Theorem 19. Let be a group and let be an element of . If for all , then .

Proof. Let . Then the induced subgraph of is complete; hence is a clique of . On the other hand, if , then there exists a subset of such that the subgraph of induced by is complete and . Note that the order of the largest clique is ; must be an element of . If , then we have being cyclic for every in . Clearly ; that is, . Let be two arbitrary elements of . So is cyclic. Since and , and are adjacent in ; yet, is adjacent to . Suppose , it is easy to see that . Since is a locally cyclic group by Definition 1, is a cyclic group; namely, is cyclic. Consequently, and are joined by an edge of . From what we have mentioned above, we can see that is a group of . Again, is a cyclic subgroup of by Definition 1. Let , where is an element of . It follows that from the hypothesis; that is, .

Corollary 20. Let be group. If , then is a clique of . Converse holds only when is the largest clique.

Corollary 21. Let . Then and are planar if and only if or .

Theorem 22. Let be a group. Then for all if and only if every element of is of prime order.

Proof. Assume that for every nonidentity element of . If there exists an element of such that is not of prime order, then we may choose such that divides the order of and . Thus and is adjacent to , since . while (otherwise, , a contradiction). so . This is contrary to .For the converse, suppose every nonidentity element of is of prime order. If is a cyclic subgroup of , then is a prime number. Thereby, , and so . That is, . Hence the theorem follows.

Theorem 23. Let be a group. Then is a cyclic subgroup for all if and only if every element of is contained in precisely one maximal cyclic subgroup of .

Proof. Assume that every element of is contained in exactly one maximal cyclic subgroup of . If is an element of , then there is a maximal cyclic subgroup such that . Let . Since is cyclic, . If , then there exists a maximal cyclic subgroup such that as . However ; this gives a contradiction to being the precisely one maximal cyclic subgroup of containing . Consequently , and so . Also, if , then , so and are adjacent in ; it means that . Thus ; that is, is cyclic. In other words, is a cyclic subgroup of .Conversely, let be an element of such that and , where and are two maximal cyclic subgroups of . Assume that is cyclic. Since is adjacent to , we have , so . Similarly, , and thus ; that is, is contained in precisely one maximal cyclic subgroup of .

Theorem 24. Let be a group. Then if and only if is isomorphic to the Klein group .

Proof. First we suppose that for group . We shall show that is isomorphic to the Klein group by the following steps.Step 1 ( is abelian). Let be an automorphism of . Then is an automorphism of group , so for all . Now we define the mapping : for all in . It is well known that is a bijection and is cyclic if and only if is cyclic; that is, is an edge of if and only if is an edge of . Thus . By hypothesis, , so for all ; namely, , and hence is a abelian group.Step 2 ( is not a cyclic group). If is a cyclic group, then we can see that is isomorphic to the complete graph by Theorem 9, and hence is isomorphic to the symmetric group . Since (if , then , but , a contradiction), is nonabelian. However, must be abelian as is cyclic, a contradiction.Step 3 ( is an elementary abelian 2-group). By Step 1, we have , where for all . It is clear that by Step 2. Obviously, there exists a graph automorphism such that , and . Since , we have and . It follows that . Furthermore, . In particular, . It follows that is an elementary abelian 2-group.Step 4 (finishing the proof). Let for some positive integer . By Step 3 and Theorem 11, is isomorphic to the star . So is the symmetric group of degree , while is isomorphic to the general linear group . Thus as . That is, .For the converse, we suppose that . Then we have . On the other hand, , and so .

Remark 25. Suppose , then . Let . Then , more specifically,
here for . Clearly, we can see that is nonabelian; that is, .

Proposition 26. Let be an elementary abelian -group for some prime integer . Then is isomorphic to Figure 3.

Figure 3: The cyclic graph of the elementary abelian -group.

Proof. Let be an element of and . Since is an elementary abelian -group, we conclude that the order of is . It follows that the subgraph of induced by is isomorphic to the complete graph of order . Let be an element such that . If and are adjacent, then is a cyclic subgroup of , which implies since is a cyclic subgroup of order 5, and this gives a contradiction to . Thus is uniquely adjacent to every vertex of . This completes the proof.

Remark 27. Let be composite. Then, in general, is not isomorphic to Figure 3. For example, let , then . In fact, .

4. The Cyclic Graphs of the Dihedral Groups

For , the dihedral group is an important example of finite groups. As is well known, . As a list,

Theorem 28. Let be the cyclic graph of and . Then(1) for any ;(2) for any ;(3) is not Eulerian;(4) is not Hamiltonian;(5) is planar if and only if or ;(6) is a split graph;(7).

Proof. (1) Clearly, the order of is 2 for all by the definition of . Since every cyclic subgroup of has a uniquely cyclic subgroup of order 2, is noncyclic; that is, and are not adjacent to each other. If is adjacent to , where , then is cyclic and hence , which is a contradiction. Thus, is the unique element of which is adjacent to , as required.(2) It is easy to see that for any . Now (1) completes the proof.(3) Since is an odd integer by (1), is not Eulerian (see [5, Theorem 6.1, page 137]).(4) In view of (1) and (2), contains a cut-vertex . In the light of [5, Theorem 6.5, page 145], we conclude that cannot be Hamiltonian.(5) If or , then it is easy to see that is planar. Now suppose that is planar. Since the complete graph of order is not planar, we have . Since the subgraph of induced by is complete, we have . That is, or , as desired.(6) By (1) and (2), the vertex set of can be partitioned into the clique and the independent set , and hence is a split graph.(7) It is straightforward.

Corollary 29. Let be the cyclic graph of and . Then is not bipartite.

Corollary 30. Let . Then .

Corollary 31. Let . Then .

Theorem 32. Let be an integer. If is a group with , then .

Proof. We have by Definition 4. In view of Theorem 28, we can see that . It follows from Theorem 19 that there exists an element such that is a cyclic subgroup of order . Note that ; we have being a normal subgroup of . Since there are vertices in such that the degrees equal 1, there exist elements of order 2 in . Now we choose an involution of order 2 of such that . It is easy to see that ; that is, . By the definition of dihedral group, acts on by inversion. This implies that , as required.

5. The Cyclic Graphs of the Generalized Quaternion Groups

The quaternion group is also an important example of finite nonabelian groups; it is given by

As a generalization of , the generalized quaternion group is defined as
where is the identity element and (if , then ). Clearly, has order as a list
Moreover, and , where .

Lemma 33. .

Proof. Since and for all , ; that is, for all . On the other hand, it is obvious that is a cyclic subgroup of for all as , where . Consequently for all , namely, . However, , so .

Proposition 34. Let be the cyclic graph of . Then(1) for all ;(2) for all and ;(3).

Proof. (1) Since for all , . Obviously, . If and are joined by an edge, then is a cyclic subgroup of order 4; Note that is a cyclic subgroup of order 4, then or . On the other hand, it is easy to see that cannot be cyclic, where and . Consequently, we have , and so .(2) By the proof of (1), we see that is not cyclic for all and , so .(3) Obviously by Lemma 33.

Corollary 35. Let . Then .

Corollary 36. Let . Then .

Theorem 37. Let be the cyclic graph of . Then is planar if and only if .

Proof. Suppose . It is easy to see that is planar. Now assume that is a planar graph. Then since is nonplanar. By Theorem 19, there exists no the element of such that . However , and hence .

Theorem 38. Let be the cyclic graph of and . Then(1) is not Eulerian;(2) is not Hamiltonian.

Proof. (1) It is similar to the proof of (3) in Theorem 28.(2) Let denote the number of components in the graph . By Theorem 6.5 of [5] on page 145, if is Hamiltonian, then for every nonempty proper set of vertices of , we have . Now suppose . Then the number of components of the resulting graph is equal to . However, , a contradiction.

6. The Cyclic Graphs of Noncyclic Groups of Order up to 14

It is significant to obtain detailed information on the cyclic graphs of some noncyclic groups of lower order. In this section, we present a table on the cyclic graphs of noncyclic groups of order up to , as shown in Table 1.

Table 1

Acknowledgments

This research is supported by the NSF of China (10961007, 11161006), and the NSF of Guangxi Zhuang Autonomous Region of China (0991101, 0991102). The authors are indebted to the anonymous referee for his/her helpful comments that have improved both the content and the presentation of the paper.